Randomly weighted sums under a wide type of dependence structure with application to conditional tail expectation

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1 Randomly weighted sums under a wide type of dependence structure with application to conditional tail expectation Shijie Wang a, Yiyu Hu a, Lianqiang Yang a, Wensheng Wang b a School of Mathematical Sciences, Anhui University, Hefei, Anhui, China, b College of Science, Hangzhou Normal University, Hangzhou, China, Abstract: Let X k, 1 k n, be n real-valued random variables, and θ k, 1 k n, be another n nonnegative not-degenerate at zero random variables. Assume that random pairs (X 1, θ 1,..., (X n, θ n are mutually independent, while each pair (X k, θ k follows a wide type of dependence structure. Consider the randomly weighted sum S θ n = n θ kx k. In this paper, the tail asymptotics for S θ n in the case where X k, 1 k n, belong to some heavy-tailed subclasses are firstly investigated. Then, as an application, we consider the tail behavior of the conditional tail expectation E(S θ n S θ n > x q as q 1, where x q = VaR q (S θ n = inf{y R : P(S θ n y q}. Under some technical conditions, the asymptotic estimate for the right tail of conditional tail expectation is also derived. The obtained results extend some existing ones in the literature. Keywords: Randomly weighted sum; asymptotic; heavy-tailed distribution; conditional tail expectation 1 Introduction Let X k, 1 k n, be n real-valued random variables (r.v.s, which are called primary r.v.s, and θ k, 1 k n, be another n nonnegative not-degenerate at zero r.v.s, which are called random weights. Consider the randomly weighted sum S θ n = θ i X i. (1.1 i=1 Tang and Yuan (2014 proposed a business model related to (1.1. In their work, assume an investor who invests in n lines of business. Each line i generates a potential net loss variable Z i in loss-profit form. Noting that the loss variables must depend on 1

2 each other since the n lines operate in similar macroeconomic environments, they modelled them by Z i = θ i X i, 1 i n. Then the randomly weighted sum (1.1 represents the total loss. Assume that each r.v. in (1.1 has a finite mean and then define the conditional tail expectation (CTE at level q (0 < q < 1 as where x q is the value at risk (VaR of S θ n, that is CTE q (Sn θ = E [ ] Sn θ S θ n > x q, (1.2 x q = VaR q (S θ n = inf{y R : P(S θ n y q}. (1.3 Clearly, the quantity CTE q (S θ n denotes the underlying risk measure of such a business model. However, the expression given by (1.2 is usually difficult to evaluate as mentioned in Tang and Yuan (2014. Suppose that X k, 1 k n, are mutually independent with distribution functions (d.f.s F k, 1 k n, respectively, and θ k, 1 k n, are nonnegative and arbitrarily dependent, but independent of X k, 1 k n. If there is a regularly-varying r.v. X distributed by F with index α > 1, denoted by F R α, such that F k (x c k F (x for some constant c k > 0 as x, then Tang and Yuan (2014 obtained an asymptotic formula for CTE as following CTE q (Sn θ α ( 1/αVaRq c k Eθk α (X, q 1, (1.4 α 1 where F R α is in the sense that F (xy y α F (x for all y > 0 and for any d.f. V, denote its right tail by V (x = 1 V (x by convention. Subsequently, following the work of Tang and Yuan (2014, Yang et al. (2015 assumed that the primary r.v.s and random weights satisfy the following two assumptions: (A1. X k, 1 k n, are real-valued r.v.s with d.f.s F k, 1 k n, respectively; and θ k, 1 k n, are nonnegative, not-degenerate at zero r.v.s with d.f.s G k, 1 k n, respectively. Moreover, random pairs (X k, θ k, 1 k n, are mutually independent. (A2. defined as For all 1 k n, random pair (X k, θ k has a bivariate Sarmanov distribution P(X k dx, θ k dy = (1 + r k ϕ k (xψ k (ydf k (xdg k (y, where the constants r k, ϕ k, ψ k satisfy some necessary conditions such that the relation defined above to be a proper distribution. 2

3 Under the assumptions (A1 and (A2, furthermore, suppose that lim inf x D Xk, x inf (1 + r k ϕ k (xψ k (y > 0, y D θk where D Xk and D θk are the support sets of X k and θ k, respectively, for every 1 k n, ψ k is uniformly continuous on D θk and there exists d k < such that lim ϕ k (x = d k. x If there is a regularly-varying r.v. X distributed by F R α with α > 1, such that F k (x c k F (x for some constant c k > 0 as x, Yang et al. (2015 proved that CTE q (Sn θ α ( 1/αVaRq c k τ k (X, q 1, (1.5 α 1 where τ k = E[θk α] + r kd k E[ψ k (θ k θk α ] for all 1 k n. It is no doubt that (1.5 is a dependent version of (1.4. Moreover, compared with (1.4, the asymptotic formula (1.5 captures the impact resulting from dependence structure. However, assumption (A2 shows that the joint distribution (X k, θ k has an explicit expression, which brings great convenience for estimating the quantity CTE. Therefore, a natural question is how to estimate the quantity CTE if the joint distribution of (X k, θ k has no explicit expression, while only understand that (X k, θ k fulfills some dependence structures. Motivated by this point, in this paper, we replace assumption (A2 with the following assumption (A3, which only addresses a dependence structure between each random pair (X k, θ k. (A3. For each fixed k = 1, 2,..., n, there exists a measurable function h k : [0, (0, such that the relation P(X k > x θ k = t h k (tp(x k > x (1.6 holds uniformly for all t 0 as x, where the uniformity is in the sense that lim sup P(X k > x θ k = t x h k (t P(X k > x = 0. t 0 When t is not a possible value of θ k, the conditional probability in (1.6 is simply understood as unconditional, and, therefore, h k (t = 1 for such t. The dependence structure in (1.6 was first proposed by Asimit and Jones (2008. Indeed, (1.6 can also be restated through copulas; see Joe (1997 or Nelsen (1998 for the concept and properties of copulas. By Sklar s theorem, for every joint distribution V (x, y with marginal distributions F (x and G(y, there is a copula function C(u, v : [0, 1] 2 [0, 1] such 3

4 that V (x, y = C(F (x, G(y, (x, y (, 2, and C is unique if F and G are continuous. Furthermore, if the copula C is absolutely continuous, then denote by c 12 = c 12 (u, v = 2 C(u, v/( u v its density. Under this framework, then the function h k (y in (1.6, if it exists, is equal to c 12 (1, G k (y. Thus, one can easily construct some concrete examples satisfying (1.6 by using the Ali-Mikhail-Haq, the Farlie-Gumbel- Morgenstern (FGM, the Frank, the Johnson-Kotz iterated FGM, the Plackett copulas; see also Asimit and Badescu (2010, Li et al. (2010 and Yang et al. (2016 for more details. In particular, for Ali-Mikhail-Haq copula C(u, v = uv, λ ( 1, 1. 1 λ(1 u(1 v By some simple calculations, one can easily check that it satisfies (A3 with a bounded measurable function h(t = 1 + λ 2λG(t, where G is the distribution of θ, while it is not included by the Sarmanov case. Based on assumption (A3, hereafter we devote ourselves to reinvestigating the asymptotic formula of CTE in this paper. In the process of doing so, we first derive some asymptotic formulas for the tail probability of the randomly weighted sum (1.1 in the case where the primary random variables belong to the intersection of dominatedly-varying-tailed and long-tailed class and the subexponentail class, respectively (see Theorems 3.1 and 3.2 below. Based on these results together with some technical conditions, then we derive an asymptotic formula for CTE as a corollary (see the following Corollary 3.1. Our results extend the works of Tang and Yuan (2014 and Yang et al. (2015. The merit of our asymptotic estimate for CTE is that it is unnecessary to assume that the joint distribution of each pair (X k, θ k has an explicit expression, while only require that (X k, θ k is asymptotically independent in some sense according to assumption (A3. The rest of this paper is organized as follows. Section 2 prepares some preliminaries of heavy-tailed distributions and lemmas. The main results and their proofs are presented in Sections 3 and 4, respectively. Section 5 gives a simulation study to verify the accuracy of the asymptotic formula of CTE. 2 Preliminaries and lemmas In this section, all limit relationships are according to x unless otherwise stated. For convenience of use, throughout this paper, for two positive functions f and g, we write f(x g(x if lim f(x/g(x = 1; write f(x = O(g(x if lim sup f(x/g(x < ; 4

5 write f(x = o(g(x if lim f(x/g(x = 0; and write f(x g(x if f and g are weakly equivalent, that is, f(x = O(g(x and g(x = O(f(x simultaneously. As usual, we write X + = X 0 and denote by 1 A the indicator function of an event A. We first present some concepts and properties of heavy-tailed subclasses for late use, then prepare some lemmas for proving our main results Heavy-tailed distributions Firstly, let us recall some important heavy-tailed subclasses. We say that a random variables X (or its distribution F is heavy-tailed, denoted by F K, if it has no finite exponential moments. One of the most important classes of heavy-tailed distributions is the subexponential distribution class. We say that a d.f. F on R + is said to be subexponential, written as F S, if F 2 (x 2F (x, where F 2 denotes the two-fold convolution of F. More generally, a d.f. F on R is still said to be subexponential if the d.f. F + (x = F (x1 (x 0 is subexponential. A d.f. F on R is said to be long-tailed, written as F L, if F (x + y F (x for all y R. A d.f. F on R is said to be dominatedly-varying tailed, denoted by F D, if F (xy = O(F (x for some 0 < y < 1. A d.f. F on R is said to be regularly-varying tailed with index α for some α 0, denoted by F R α, if for each y > 0, F (xy y α F (x. Let R be the union of R α over the range 0 α <. Then the following inclusions are proper: R D L S L K. See Embrechts et al. (1997 for these well-known results. Besides, we say a d.f. F on R is said to belong to the class A, which was first proposed by Konstantinides et al. (2002, 5

6 if it is subexponential and lim sup x F (xy F (x < 1 for some y > 1. (2.1 Note that (2.1 is a mild condition fulfilled by almost all useful distributions with an ultimate right tail, see Tang (2006 or Tang and Yuan (2014 for more details. For some other subclasses of heavy-tailed distributions and their properties, we refer the reader to Embrechts et al. (1997 and the references therein. where Moreover, for a d.f. F with an ultimate right tail, define { } J + F = inf log F (y : y > 1, log y F (y = lim inf x F (xy F (x. In the terminology of Tang and Tsitsiashvili (2003, J + F is called the upper Matuszewska index of F. By definition, one can easily see that J + F = α if F R α. Moreover, from Proposition of Bingham et al. (1987,it holds that x p = o(f (x (2.2 holds for every p > J + F. If F D, then for every p > J + F, there are some positive constants K and x 0 such that the inequality holds for all xy x x 0. F (x F (xy Kyp ( Some lemmas In this subsection, we prepare some lemmas for the proof our main results. Lemma 2.1 below describes the subexponentiality for the product of two dependent r.v.s, which is due to Wang et al. (2017. It is a direct extension of Theorem 2.1 of Tang (2006 from independent case to dependent case. 6

7 Lemma 2.1. Let X be a real-valued r.v. distributed by F and θ be a nonnegative, not-degenerate at zero r.v.. Moreover, assume that there exists a measurable function h such that the relation holds uniformly for all t [0,. If F A, 0 < relation holds for each u > 0, then θx A. P(X > x θ = t h(tp(x > x (2.4 inf h(y sup h(y < and the y [0, y [0, P(θ > ux = o(1p(θx > x (2.5 The following Lemma 2.2 is a slight adjustment of Lemma 2.4 of Yang et al. (2014 which extends the famous Breiman lemma (see Breiman, 1965 and plays a key role in the proof of our result. Lemma 2.2. Let X be a real-valued r.v. distributed by F R α (α 0 and θ be a nonnegative, not-degenerate at zero r.v.. Moreover, assume that X and θ are dependent according to (2.4. If P(θ > x = o(f (x and E[θ p h(θ] < for some p > α, then P(θX > x E[θ α h(θ]f (x. The last Lemma is due to Resnick (1987, see Proposition 0.8 (vi which describes a relation between the asymptotic tail behavior of regularly varying distribution functions and the behavior of corresponding quantile functions in the neighborhood of unity. Lemma 2.3. Let ξ and η be two r.v.s such that F ξ, F η R α for some α > 0. Then, for some positive c, F η (x cf ξ (x if and only if VaR q (η c 1/α VaR q (ξ, q 1. 3 Main results In this section we present our main results. Theorems 3.1 and 3.2 derive some asymptotic formulas for the tail probabilities of the randomly weighted sums and their maxima in the case where the primary r.v.s belong to the heavy-tailed subclass of D L and A, respectively. Finally, the asymptotic formula for CTE of randomly weighted sums with regularly-varying primary r.v.s is given as a corollary. 7

8 Theorem 3.1. Suppose that random pairs (X k, θ k, 1 k n, satisfy (A1 and (A3. If, for all k = 1,..., n, F k D L and max{eθ p k k, E[θp k k h(θ k]} < for some p k > J + F k, then ( ( P max 1 i n Sθ i > x P(Sn θ > x P θ k X + k > x P(θ i X i > x. (3.1 i=1 Theorem 3.2. Suppose that random pairs (X k, θ k, 1 k n, satisfy (A1 and (A3 with 0 < inf h k(y sup h k (y < for all k = 1,..., n. If, for all 1 k n, y [0, y [0, F k L, F k (x F (x for some F A and the relation holds for all u > 0, then the relations in (3.1 hold. Corollary 3.1. P(θ k > ux = o(1p(θ k X k > x, (3.2 Suppose that random pairs (X k, θ k, 1 k n, satisfy (A1 and (A3. Furthermore, assume that there exists a regularly-varying r.v. X distributed by F R α with α > 1, and positive constants c 1,..., c n such that F k (x c k F (x for all k = 1,..., n. If max{e[θ p k ], E[θp k h(θ k]} < for some p > α and all k = 1,..., n, then we have that CTE q ( S θ n α ( α 1 i=1 c i E[θ α i h i (θ i ] 1 α VaR q (X, q 1. (3.3 Clearly, Theorems 3.1 and 3.2 coincide with Theorems 2 and 3 of Tang and Yuan (2014. Besides, in assumption (A2, if we further assume that ψ k is uniformly continuous on D θk, and there exists d k < such that lim x ϕ k (x = d k, Lemma 3.5 of Yang et al. (2015 implies that (X k, θ k fulfills assumption (A3. Therefore, from this point of view, Corollary 3.1 is a substantial extension of Theorem 2.2 in Yang et al. (2015. Moreover, the merit of our Corollary 3.1 is that it is unnecessary to assume that the joint distribution of each pair (X k, θ k has an explicit expression, while only require that each random pair (X k, θ k is dependent according to assumption (A3. 4 The proofs of main results 4.1. Proof of Theorem 3.1 Firstly, by Markov s inequality and (2.2, it holds that P(θ k > x x p k Eθ p k k = o(1f k (x. It follows from Lemma 1 of Yang et al. (2013 that θ k X k belongs to the class 8

9 L for all k = 1,..., n. Thus, we can choose some positive function l(x, with l(x and l(x x/2, such that the relation P(θ k X k > x+y P(θ k X k > x for all l(x y l(x and k = 1,..., n. Following the proof idea of Theorem 3 of Tang and Yuan (2014, we first start to estimate the asymptotically lower bound of the tail probability of S θ n. By Bonferroni s inequality, it holds that P ( Sn θ > x ( P Sn θ > x, n P ( Sn θ > x, θ k X k > x + l(x = (1 + o(1 (1 + o(1 = (1 + o(1 P(θ k X k > x + l(x P(θ k X k > x + l(x P(θ k X k > x + l(x P(θ k X k > x, θ k X k > x + l(x 1 j<k n P(θ j X j > x + l(x, θ k X k > x + l(x ( P Sn θ x, θ k X k > x + l(x ( P θ k X k > x + l(x, j=1,j k ( ( P θ k X k > x + l(x P θ j X j < l(x j=1,j k θ j X j < l(x where in the third and last steps we used the fact that random pairs (X k, θ k, 1 k n, are mutually independent. For the asymptotically upper bound, we have that ( P θ k X + k > x = ( n P θ k X k > x l(x + P P(θ k X k > x l(x + P(θ k X k > x l(x + ( θ k X + k n > x, θ k X k x l(x, P (θ j X j > x n, j=1 j=1,k j ( P θ j X j > x ( P n,k j θ k X + k > l(x n θ j X j > x n j=1 θ k X + k > l(x P(θ k X k > x, (4.2 where in the last step we used the fact that θ j X j D by applying Lemma 2 of Yang et al. (2013 and thus P (θ j X j > x/n P (θ j X j > x. Combining (4.1 and (4.2 together 9 (4.1

10 with the following fundamental inequality ( ( ( P Sn θ > x P max 1 i n Sθ i > x P θ k X + k > x yields the desired (3.1. This ends the proof of Theorem Proof of Theorem 3.2 Under the conditions of Theorem 3.2, it follows from Lemma 2.1 that θ k X k belongs to the class A for all k = 1,..., n. Note that θ 1 X 1,..., θ n X n are mutually independent and A S. Thus, Theorem 3.2 is directly derived from Theorem 3.1 of Tang and Tsitsiashvili (2003 with all weights equalling to Proof of Corollary 3.1 In order to prove (3.3, we first prove that for each fixed l {1,..., n}, E ( θ l X l S θ n > x q α α 1 ( n c l E[θl αh l(θ l ] 1 1 VaR q (X, q 1. (4.3 i=1 c ie[θi αh α i(θ i ] Note that Eθ p k < for all k = 1,..., n. By Markov s inequality and (2.2, it holds that P(θ i > x x p Eθ p i = o(p(x i > x. Hence, it follows from Theorem 3.1 and Lemma 2.2 that P ( S θ n > x P(θ i X i > x F (x i=1 c i E[θi α h i (θ i ], x. (4.4 For any fixed l {1,..., n}, directly applying Theorem 2.1 of Yang et al. (2015 and integration by parts yields that Eθ l X l 1 {S θ lim n >x} x xf (x i=1 = lim x Eθ l X l 1 {θl X l >x} xf (x xp(θ l X l > x + x = lim x xf (x ( 1 + = lim x P(θ l X l > x F (x = lim udp(θ x lx l > u x xf (x P(θ 1 l X l > xydy 1 P(θ l X l > xydy P(θ l X l > x. (4.5 By Theorem of Bingham et al. (1987, the Potter s bounds for regularly varying distributions are known, that is, for any d.f. V R α, ε (0, 1, y > 0 and sufficiently large x, it holds that (1 ε ( y α+ε y α ε V (xy V (x (1 + ε( y α+ε y α ε. 10

11 Note that F R α with α > 1. According to relation (4.4, θ l X l is also regularly varying with the same index. Arbitrarily choosing 0 < ε < (α 1 1, for y > 1, the upper Potter s bound of θ l X l implies that P(θ l X l > xy P(θ l X l > x (1 + εy α+ε. Thus, it follows from the dominated convergence theorem that P(θ 1 l X l > xydy lim = y α dy = 1 x P(θ l X l > x α 1. (4.6 By (4.5 and (4.6, applying Lemma 2.2 again yields that Eθ l X l 1 {S θ lim n >x} x P(Sn θ > x 1 = α c l E[θl αh l(θ l ] α 1 n i=1 c ie[θi αh i(θ i ] x. Choosing x = VaR q (S θ n = F S θ n(q in above relation, we conclude that Eθ l X l 1 {S θ n >x} P(S θ n > x α c l E[θl αh l(θ l ] α 1 n i=1 c ie[θi αh i(θ i ] F S (q, q 1, (4.7 n θ where F S θ n(q is the generalized converse function of distribution function F S θ n (x = P(S θ n x. Note that F R α with α > 1. According to the first relation of (4.4, F S θ n is also regularly varying with the same index. By the second relation of (4.4 and Lemma 2.3, we have that F S θ n (q ( i=1 1 c i E[θi α α h i (θ i ] VaRq (X, q 1. (4.8 (4.8 together with (4.7 implies (4.1. Taking summation on both sides of (4.3 yields the desired (3.3. This ends the proof of Corollary Simulation In this section, we perform a simulation study on the result derived in Corollary 3.1. Assume that there are three types of risks {X k, k = 1, 2, 3} which are identically distributed with a common Pareto distribution F (x = 1 (1 + x α, x 0, 11

12 and the random weights {θ k, k = 1, 2, 3} are also identically distributed with a common uniform distribution on [0, 1]. The dependence structure between X k and θ k, (k = 1, 2, 3 is described by the following FGM copula C(u, v = uv + ruv(1 u(1 v, r [ 1, 1]. For convenience, denote by G the distribution of θ. Hence, the function h in relation (3.3 is equal to h(t = 1 + r(1 2G(t. In our simulation, the parameter α is chosen to be 2, 3 and 4, and r is chosen to be 0.5, 0 and 0.5, respectively. Our simulation is based on three confidence levels: q = 0.95, 0.97 and Each analysis is preformed for 1000 samples consisting of simulations from (θ k, X k. To verify the accuracy of the results of Corollary 3.1, here we characterize the convergence rate in Corollary 3.1 by the value of the left-hand side of relation (3.3 divided by the right-hand side of relation (3.3. It is expected that the value defined above is close to 1. Our simulation results are presented in the following Table 1. r = 0.5 r = 0 r = 0.5 q α = α = α = Table 1: Convergence rate of relation (3.3 Table 1 shows that, generally, heavier tails (smaller values for α entail faster convergence of the rates to 1 and the rates mainly increase for large values of q as expected. However, the convergence rate increases less obviously as the strength of dependence relaxes (closer values to zero for r. Acknowledgements The authors would like to thank the anonymous referees for their insightful and constructive suggestions which have helped us significantly improve the paper. The authors 12

13 would also like to thank Dr. Hongyan Fang for providing the simulations. This work was supported by the National Natural Science Foundation of China ( , , the Provincial Natural Science Research Project of Anhui Colleges (KJ2017A024, KJ2017A028. References [1] Asimit, A.V., Badescu, A.L., Extremes on the discounted aggregate claims in a time dependent risk model. Scand. Actuar. J. 2, [2] Asimit, A.V., Jones, B.L., Dependence and the asymptotic behavior of large claims reinsurance. Insur. Math. Econ. 43(3, [3] Bingham, N.H., Goldie, C.M., Teugels, J.L., Regular Variation. Cambridge University Press, Cambridge. [4] Breiman, L., On some limit theorems similar to the arc-sin law. Teor. Verojatnostej Primenenjia 10, [5] Embrechts, P., Klüppelberg, C., Mikosch, T., Modelling Extremal Events for Insurance and Finance. Springer, Berlin, Heidelberg. [6] Joe, H., Multivariate models and dependence concepts. London: Chapman Hall. [7] Konstantinides, D., Tang, Q., Tsitsiashvili, G., (2002. Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails. Insur. Math. Econ. 31, [8] Li, J., Tang, Q., Wu, R., Subexponential tails of discounted aggregate claims in a timedependent renewal risk model. Adv. Appl. Probab. 42, [9] Nelsen, R. B., An introduction to copulas. New York: Springer. [10] Resnick, S.I., Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, New York. [11] Tang, Q., The subexponentiality of products revisited. Extremes 9(3-4, 231õ241. [12] Tang, Q., Tsitsiashvili, G., Randomly weighted sums of subexponential random variables with application to ruin theory. Extremes 6, [13] Tang, Q., Yuan, Z., Randomly weighted sums of subexponential random variables with application to capital allocation. Extremes 3, [14] Wang, S., Zhang, C., Wang, X., Wang, W., The finite-time ruin probability of a discrete-time risk model with subexponential and dependent insurance and financial risks. Acta Math. Appl. Sin. (Engl. Ser.. Preprint. [15] Yang, H., Gao, W., Li, J., Asymptotic ruin probabilities for a discrete-time risk model with dependent insurance and financial risks. Scand. Actuar. J. 1, [16] Yang, Y., Ignatavičiūtė, E., Šiaulys. J., Conditional tail expectation of randomly weighted sums with heavy-tailed distributions. Stat. Probab. Lett. 105, [17] Yang, Y., Leipus, R., Šiaulys, J., Tail probability of randomly weighted sums of subexponential random variables under a dependence structure. Stat. Probab. Lett. 82, [18] Yang, Y., Leipus, R., Šiaulys, J., A note on the max-sum equivalence of randomly weighted sums of heavy-tailed random variables. Nonlinear Anal. Model. Control 18(4,